Database System Concepts, 5 th Ed. Silberschatz, Korth and Sudarshan See for conditions on re- Chapter 2: Relational

Database System Concepts, 5th Ed.©Silberschatz, Korth and Sudarshan

See www.db-book.com for conditions on re-use

Chapter 2: Relational Model

http://www.db-book.com/

©Silberschatz, Korth and Sudarshan2.2Database System Concepts - 5th Edition, Oct 5, 2006

ExamplesConsider the following relations: Student(roll, name, address, phone) Course(code, title) Registered(roll, code)

• Find out the course code in which at least one student is registeredπcode ( Registered)

• Find out the titles of registered courses πtitle

(Course.code = Registered.code (Course X Registered ))

• Find out the course code in which no student is registered πcode

( Course ) - πcode ( Registered )


Examples

• Find out the student names and course titles they registered toπname, title

(Student.roll = Registered.roll^ Registered.code=Course.code (Student X Registered X Course))

Consider the following relations: Student(roll, name, address, phone) Course(code, title) Registered(roll, code)

• Name of student who are registered to ‘Database’ or ‘Algorithms’ course πname(Student.roll = Registered.roll^ Registered.code=Course.code(Student X Registered X (σ title=’Database’

Course)) ) ∪πname(Student.roll = Registered.roll^ Registered.code=Course.code(Student X Registered X (σ title=’Algorithm’

Course)) )


Rename Operation Allows us to name and therefore to refer to the results of relational-

algebra expressions. Allows us to refer to a relation by more than one name. Example:

x (E)

returns the expression E under the name X If a relational-algebra expression E has arity n, then

returns the result of expression E under the name X, and with theattributes renamed to A1 , A2 , …., An .

)(),...,,( 21E

nAAAx


Banking Examplebranch (branch_name, branch_city, assets)

customer (customer_name, customer_street, customer_city)

account (account_number, branch_name, balance)

loan (loan_number, branch_name, amount)

depositor (customer_name, account_number)

borrower (customer_name, loan_number)


Example Queries Find the largest account balance

Strategy: Find those balances that are not the largest

– Rename account relation as d so that we can compare each account balance with all others

Use set difference to find those account balances that were not found in the earlier step.

The query is:

balance(account) - account.balance

(account.balance < d.balance (account x d (account)))


Example Queries Find the names of all customers who have a loan at the Dhanmondi

branch.

Query 2

customer_name(loan.loan_number = borrower.loan_number (

(branch_name = “Dhanmondi” (loan)) x borrower))

Query 1

customer_name (branch_name = “Dhanmondi” (

borrower.loan_number = loan.loan_number (borrower x loan)))


Formal Definition

A basic expression in the relational algebra consists of either one of the following: A relation in the database A constant relation

Let E1 and E2 be relational-algebra expressions; the following are all relational-algebra expressions:

E1 E2

E1 – E2

E1 x E2

p (E1), P is a predicate on attributes in E1

s(E1), S is a list consisting of some of the attributes in E1

x (E1), x is the new name for the result of E1


Additional Operations

We define additional operations that do not add any power to therelational algebra, but that simplify common queries.

Set intersection Natural join Division Assignment


Set-Intersection Operation Notation: r s Defined as: r s = { t | t r and t s } Assume:

r, s have the same arity attributes of r and s are compatible

Note: r s = r – (r – s)


Set-Intersection Operation – Example

Relation r, s:

r s

A B

121

A B

23

r s

A B

2


Example Query Find the names of all customers who have a loan and an account at

the bank.

customer_name (borrower) customer_name (depositor)


Notation: r s

Natural-Join Operation

Let r and s be relations on schemas R and S respectively. Then, r s is a relation on schema R S obtained as follows:

Consider each pair of tuples tr from r and ts from s.

If tr and ts have the same value on each of the attributes in R S, add a tuple t to the result, where

t has the same value as tr on r

t has the same value as ts on s

Example:R = (A, B, C, D)S = (E, B, D) Result schema = (A, B, C, D, E) r s is defined as:

r.A, r.B, r.C, r.D, s.E (r.B = s.B r.D = s.D (r x s))


Natural Join Operation – Example Relations r, s:

A B

12412

C D

aabab

B

13123

D

aaabb

E

r

A B

11112

C D

aaaab

E

s

r s


Example Query

Find the name and loan amount of all customers who have a loan at the bank.

customer_name, amount (borrower loan)

customer. customer_name (customer_city = “Jessore” (customer depositor account ))

Find the names of all customers who have an account in the bank and who live in Jessore

Natural Join is associative


Example Query Find the names of all customers who have a loan and an account at

the bank.

customer_name (borrower) customer_name (depositor)

customer_name (borrower depositor)

When R S =R X S ?

If R S= φ


Summary


Division Operation

Notation: Suited to queries that include the phrase “for all”.

Let r and s be relations on schemas R and S respectively where R = (A1, …, Am , B1, …, Bn )

S = (B1, …, Bn)

The result of r s is a relation on schema

R – S = (A1, …, Am)

r s = { t | t R-S (r) u s ( tu r ) }

Where tu means the concatenation of tuples t and u to produce a single tuple

r s


Division Operation – Example

Relations r, s:

r s:

B

A

1

2

A B

12311134612

r

s


Another Division Example

A B

aaaaaaaa

C D

aabababb

E

11113111

Relations r, s:

r s:

D

ab

E

11

A B

aa

C

r

s


Find all customers who have an account at all branches located in Brooklyn city.

Bank Example Queries

customer_name, branch_name (depositor account)

branch_name (branch_city = “Brooklyn” (branch))


Assignment Operation

The assignment operation () provides a convenient way to express complex queries. Write query as a sequential program consisting of

a series of assignments followed by an expression whose value is displayed as a result of

the query. Assignment must always be made to a temporary relation variable.

Example:

The result to the right of the is assigned to the relation variable on the left of the .

May use variable in subsequent expressions.

r1 branch_city = “dhaka” (account branch )

r2 account_number (r1)

Documents

Database System Concepts, 5 th Ed. Silberschatz, Korth and Sudarshan See for conditions on re- Chapter 2: Relational